If the grout in between tiles in the shower or other damp environment is not properly sealed, it can be porous. This can cause mold and mildew to grow and make it both unsightly and unhealthy to be around. The best situation is to have the grout properly sealed when it is applied and to reseal it periodically after a good cleaning. This will help prevent the growth of mold and mildew. You can get grout sealant at any home supply store.
To clean grout between tiles, you can use a spray on tub and tile cleaner, or you can use straight bleach. Either way, be sure to have good ventilation because it is dangerous to breathe the fumes. If you use a spray on, simply rinse the surface lightly then spray the product over the entire surface of the grout and the tiles. Allow it to sit for half an hour or so. Scrub the entire surface with a medium bristle scrub brush and rinse with hot water. Address any remaining spots with a small, stiff bristled brush and straight bleach.
If you do not want to spray toxic chemicals over a large surface, you can paint straight bleach onto the grout only using a small paint brush. You still need to ventilate well, but the fumes won’t be as pervasive. Allow the bleach to sit for half an hour or so then rinse with hot water.
After rinsing, in either case, put a fan on the area and allow it to dry thoroughly. Inspect for areas of grout that need mending and repair them as needed. Seal the grout using any brand of grout sealant and a small artists paintbrush. Be sure to allow your work to dry thoroughly between each step.